The overarching goal of this proposal is to apply and develop a class of stochastic models known as general branching processes for various biological/biomedical applications that involve cellular population dynamics. These models will be used to explain data, estimate parameters, assess specific hypotheses, and make predictions. Specific projects are: (1) asynchronicity in cell populations, relevant to cancer therapy;(2) telomere dynamics, relevant to processes of aging and also to cancer therapy;(3) bacterial lag phase estimation, relevant to food microbiology;and three projects relevant to degenerative genetic disease: (4) accumulation of deleterious mutations in mitochondrial DNA, (5) cell populations with conflicting levels of selection where one type of DNA may be favored inside the cell but not for selection among cells, and (6) the "ratchet" process of accumulation of harmful mutations. There is a great need for mathematical modeling and quantitative data analysis in biology in general, and in cellular population dynamics in particular. Although different mathematical approaches have been taken to many of these problems, branching processes seem to be able to offer great improvement in that they are inherently stochastic, taking into account the natural randomness in cell cycle times, mutations, etc. and that they are conceptually clear, starting from modeling behavior on the individual level, for example by modeling the cell cycle and estimating relevant parameters. The proposed research is intended to develop models and estimation techniques that are biologically sound and improve accuracy in computation and estimation. 1 PUBLIC HEALTH RELEVANCE: The biomedical problems that are here proposed to be addressed with mathematical methods are all relevant to public health. The problem of desynchronization of cell populations is relevant to cancer therapy, the problem of shortening of telomeres is relevant to processes of aging and also to cancer therapy, the problem of bacterial lag phase estimation is relevant to food safety, and the problems of accumulation of mutations is relevant to genetic disease. The proposed mathematical methods (general branching processes) are conjectured to improve modeling and estimation in the cellular population processes involved in these problems. 1